The Role of a Simple Inerter in Seismic Base Isolation

Abstract

The present study investigates the role of a simple inerter in supplemental devices for possible implementation in the mature seismic base isolation technique. Firstly, the response of the base-isolated structure with an optimally tuned mass damper inerter (TMDI) is investigated to see the tuning effects. The time required to tune the TMDI was found to be significantly longer than the duration of a strong-motion earthquake. There was still a reduction in the response of the isolated structure, which is primarily due to the added damping and stiffness (ADAS) of TMDI and not because of the tuning effects. Hence, it is proposed that the corresponding ADAS of the TMDI be directly added to the isolation device. Secondly, the response of the base-isolated structures to the fluid inerter damper (FID) is studied. It was observed that the inerter of the FID does not influence the displacement variance of an isolated structure under broadband earthquake excitation. It implies that the response of the isolated structure to FID is primarily controlled by its counterpart fluid damper (FD). The performance of optimal TMDI, ADAS, FID, and FD to mitigate the seismic response of the flexible multi-story base-isolated structure under real earthquake excitations is also investigated. In terms of suppressing the displacement and acceleration responses of the isolated structure, it has been found that TMDI and ADAS perform similarly. Comparing the response of the isolated structure with FID and FD demonstrated that the inerter in the FID has detrimental effects on the isolated structures, in which the top floor’s acceleration and base shear are substantially increased.

1. Introduction

Inerter-based dampers have recently seen tremendous growth in popularity for controlling structural vibration [1]. An inerter is used for mass augmentation effects, and its resisting force is predicted to be proportional to the relative acceleration between ends [2]. A tuned mass damper inerter (TMDI) simulates a conventional tuned mass damper (TMD) with an inerter. Additionally, it is referred to as a tuned inerter damper (TID) when the TMD mass is replaced with the inerter. The performance of base-isolated structures with TMDI fixed to the isolation floor under both real and stochastic earthquake excitation was examined [3,4,5,6,7,8,9,10,11,12,13]. It was demonstrated that the TMDI reduces the response of base-isolated structures equipped with linear as well as non-linear isolators. An enhanced performance of the optimal TMDI was further observed when it was placed on the upper floors of the isolated structure or used in tandem [14,15]. Recently, it has also been demonstrated experimentally that it is possible to achieve reasonable results in the response of base-isolated structures augmented with TMDI by using a simplified linear dissipative model instead of a nonlinear model with friction and gear backlash [16,17]. The efficiency and optimum TID parameters for the base-isolated structure when considering the stochastic model of earthquake excitation are also illustrated [18,19,20,21,22,23]. To reduce the seismic response and vibration isolation in the base-isolated structures, a geometrically nonlinear TID arrangement is also being investigated for multidirectional isolation [24]. The use of multiple TMDs to control the seismic response of base-isolated buildings has recently been studied [25,26]. In seismic base isolation, the supplemental fluid inerter damper (FID) and electromagnetic inertial mass damper (EIMD) represent novel applications, with the FID incorporating fluid inerter technology and the EIMD integrating electromagnetic inertial mass principles to enhance seismic resilience. The performance of the base-isolated structures with FID has been investigated by various researchers and has shown that these are effective in seismic response control [27,28,29,30,31,32,33,34]. Lately, there has been a showcase of the dynamic characteristics of electromagnetic inertial mass dampers (EIMDs) and the identification of optimal parameters to effectively control the structural response of base-isolated structures [35,36]. The review above suggests that the researchers are very interested in using inerter-based supplemental devices for base-isolated structures.

The period of base-isolated structures is usually quite long, and the tuning effects of TMDI, TID, or TMD may not be achieved during the strong motion duration of the earthquake excitation. However, there is still a reduction in the response of the isolated structure, which is primarily due to the added damping and stiffness (ADAS) of the above devices and not because of the tuning effects [37]. The tuned damper-type devices are relatively less effective for the system with higher damping than the corresponding systems with lower damping [38]. It implies that for base-isolated buildings generally equipped with heavy damping, the supplemental tuned mass-type damper devices may not be very effective. Further, in the FID and EIMD, the inerter works in tandem with the fluid damper (FD) and electromagnetic damper (ED), and it’s unclear whether the inerter or the FD and ED provide beneficial effects in response reduction. It may be possible that a single FD or ED can achieve similar or even better beneficial effects. As a result, it will be interesting to investigate the controlling effects of the inerter by examining the seismic response of base-isolated structures with FID or EIMD. This is proposed to be achieved by comparing the response of isolated structures with FID or EIMD with the corresponding response without inerter in the FID or EIMD devices.

Herein, the response of the base-isolated structures with supplemental optimal TMDI and FID is investigated for the tuning effects of TMDI and the role of inerter in FID. The current study focuses on the following specific objectives: (i) to explore the tuning effects of TMDI for seismic response control of the base-isolated structure equipped with TMDI; (ii) to compare and contrast the seismic response of base-isolated structures with TMDI to the corresponding response with the ADAS device; (iii) to understand the role of inerter in FID for seismic response control of the base-isolated structure by comparing the corresponding response with FD; and (iv) to study the comparative performance of TMDI, ADAS, FID, and FD devices for flexible multi-story base-isolated structures subjected to real earthquakes.

2. Base-Isolated Rigid Structure with TMDI and ADAS

The multi-story building model in Figure 1a is thought of as having a rigid superstructure, base isolation, and TMDI or ADAS. The base mass and the superstructure’s combined mass are represented by the mass m. The equivalent linear force–deformation behavior with viscous damping, which is specific to the building, defines the isolation system. Let k_b and c_b represent the equivalent stiffness and damping of the selected isolation system, respectively. Two parameters, namely, the isolation period T_b and damping ratio ${\xi }_b$, are used to define the base isolation system under study and are expressed as

$$T_b=\frac{2\pi }{{\omega }_b}, {\omega }_b=\sqrt{\frac{k_b}{m}} \mathrm{and} 2{\xi }_b{\omega }_b=\frac{c_b}{m}$$

(1)

2.1. Supplemental TMDI Device

The tuned mass damper inerter (TMDI) device comprises components including an auxiliary mass (m_t), stiffness (k_t), and damper (c_t), collectively known as TMD, along with an inertial device featuring an inertance coefficient denoted b (refer to Figure 1b). The two terminals of the inertial device are connected to the auxiliary mass and the ground. A reaction force is generated by the inertial device’s motion that, roughly speaking, can be regarded as proportional to the relative acceleration developed between the two terminals. The inertance, also known as apparent mass, is a proportional coefficient that can be numerous times the inerter’s actual mass. The energy is dissipated through damping when the resonance in the TMD is introduced together by the vibration of the base-isolated structure. The modeling aspects of the TMDI for practical usage had been confirmed through an experimental study of the dynamic response of an isolated structure system equipped with TMDI tested on a shaking table [16,17]. Also, TMDI is observed to be robust when a change in the design parameters from their optimal values does not produce significant response variations [39]. The auxiliary and inertial masses coupled to the TMDI are represented by the mass and inertance ratio, defined as

$$\mu =\frac{m_t}{m} \mathrm{and} \beta =\frac{b}{m}$$

(2)

The TMDI’s corresponding stiffness and damping are expressed as

$${\xi }_t=\frac{c_t}{2(m_t+b){\omega }_t}, {\omega }_t=\sqrt{\frac{k_t}{m_t+b}} \mathrm{and} f=\frac{{\omega }_t}{{\omega }_b}$$

(3)

where ${\xi }_t$ and f represent the damping ratio of the TMDI and tuning frequency ratio, respectively.

The equations governing the motion of the building model with a rigid superstructure, base isolation, and TMDI are represented by

$$\left[\begin{array}{cc}m& 0\\ 0& m_t+b\end{array}\right]\left\{\begin{array}{c}\ddot{x}\\ {\ddot{x}}_t\end{array}\right\}+\left[\begin{array}{cc}{c}_b+c_t& -c_t\\ -c_t& c_t\end{array}\right]\left\{\begin{array}{c}\dot{x}\\ {\dot{x}}_t\end{array}\right\}+\left[\begin{array}{cc}{k}_b+k_t& -k_t\\ -k_t& k_t\end{array}\right]\left\{\begin{array}{c}x\\ x_t\end{array}\right\}=-\left\{\begin{array}{c}m\\ m_t\end{array}\right\}({\ddot{x}}_g)$$

(4)

where x = relative displacement of the isolated structure; x_t = relative displacement of the auxiliary mass of the TMDI; and ${\ddot{x}}_g$ = earthquake acceleration.

Let the earthquake acceleration be harmonic as ${\ddot{x}}_g=e^{j\omega t}$; where ω is the circular frequency and $j=\sqrt{-1}$, the steady-state displacement x is expressed as

$$x=H_x\left(j\omega \right)\hspace{0.33em}{e}^{j\omega t}$$

(5)

where $H_x\left(j\omega \right)$ = frequency response function (FRF) of the displacement x.

The SDOF model of the base-isolated structure with TMDI is subjected to white-noise earthquake acceleration with the power spectral density function (PSDF) as S₀. The mean square response of base displacement for the isolated structure is given by the following equation:

$${\sigma }_x^2={{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left|H_x\left(j\omega \right)\right|}^2{S}_{0}d\omega$$

(6)

By dividing the corresponding response from the base-isolated structure with no control, the mean square response obtained from Equation (6) is further normalized, i.e.,

$${\tilde{\sigma }}_x^2=\frac{{\sigma }_x^2}{{\sigma }_{x,0}^2} \mathrm{and} {\sigma }_{x,0}^2=\frac{\pi S_0}{2{\xi }_b{\omega }_b^3}$$

(7)

The TMDI system is effective at response mitigation if the normalized variance, ${\tilde{\sigma }}_x^2$ value, is less than unity.

The parameters $\mu$, $\beta$, ${\xi }_t$, and f provide a complete description of the supplemental TMDI for the base-isolated structure. For all numerical solutions in the current study, μ is taken to be equal to 0.01 because the TMDI performs best for a smaller value of μ and higher values of β [39]. It is also worth mentioning that when μ = 0, the TMDI is reduced to the TID. When ${\xi }_t$ and f are set to their optimum values, the TMDI performs at its best. For a given base-isolated structure (i.e., specified by T_b and ${\xi }_b$) and white-noise earthquake excitation, the optimum parameters of the TMDI (i.e., optimum ${\xi }_t$ and f) are the ones that minimize the ${\sigma }_x^2$. This is referred to as ‘optimal TMDI’ in the present study and the corresponding optimums ${\xi }_t$ and f are obtained using an algorithmic numerical search technique for values spanning their feasible regions with increments of 0.0001. Using the above-specified procedure with isolation damping, ${\xi }_b$ not exceeding 10 percent, the optimum damping ratio (${\mathsf{\xi }}_t^{opt}$) and tuning frequency ratio ($f^{opt}$) for a selected value of $\mathsf{\beta }$ can be calculated using the following expressions from [40].

$${\mathsf{\xi }}_t^{opt}= \sqrt{\frac{\mathsf{\beta }(4+3\mathsf{\beta })}{8(1+\mathsf{\beta })(2+\mathsf{\beta })}}$$

(8)

$$f^{opt}=\frac{1}{1+\mathsf{\beta }}\sqrt{\frac{(1+\frac{\mathsf{\beta }}{2})}{(1+{\mathsf{\xi }}_b)}}$$

(9)

The changes in FRF of base displacement for an isolated structure with optimal TMDI are plotted in Figure 2. The inertance ratio is considered to be 0.1, 0.2, and 0.4, and the FRF responses are compared with the corresponding base-isolated structure without TMDI (referred to as BIS). The FRF of isolated structures with optimal TMDI appears to be a two-peaked response. The optimization of parameters occurs when the response to the second peak is minimized. The maximum reduction in FRF occurs at the resonating frequency, and there is a reduction of 60.8, 68.1, and 75.1 percent for the inertance ratio of 0.1, 0.2, and 0.4, respectively. Thus, it can be concluded that the TMDI effectively reduces the FRF and, thereby, the mean square of the displacement response of the base-isolated structure.

It was observed in Figure 2 that the optimal TMDI mitigates the FRF of the base-isolated structure under resonating conditions. However, it is interesting to know how much time or the number of cycles the TMDI needs to attain fully tuned conditions for maximum response reduction. To verify this, the displacement of the base-isolated structure with and without optimal TMDI subjected to earthquake acceleration of ${\ddot{x}}_g=\mathit{sin}(2\pi t/3)$ m/s² (t denotes here the time in s) is calculated using the numerical integration procedure and plotted in Figure 3. It is observed from the figure that after one cycle of response, the reduction of the displacement is 12.4, 27.7, and 39.1 percent for the inertance ratio of 0.1, 0.2, and 0.4, respectively. To achieve the maximum reduction in the response under resonating conditions, the TMDI should undergo several cycles of responses (i.e., about 10 cycles as per Figure 3). Normally, such many cycles may not be possible during the strong motion duration of past recorded earthquakes. Despite that, there is a reduction in the response of the TMDI in the initial cycles, which is primarily attributed to the added heavy damping of its auxiliary damper. It is to be noted that the added damping by optimal TMDI is 0.16, 0.48, and 0.87 times the isolation damping for the inertance ratio of 0.1, 0.2, and 0.4, respectively.

2.2. Supplemental ADAS Device

Figure 3 shows that the ADAS is responsible for the decrease in displacement of the base-isolated structure with optimal TMDI in the first cycles [37]. It is to be further noted that TMDI in base-isolated structures will be ineffective when subjected to near-fault pulse-type earthquake ground motions [41]. Under this condition, the response of the isolated structure is governed by the pulses associated with the earthquake record, and the tuning effects of TMDI will be unachievable. In view of the above facts, in place of the TMDI, it is proposed that the ADAS be added parallel to the isolation system, which may provide better control (refer to Figure 1c). For a specified inertance, the corresponding stiffness (k_t), and damper (c_t) of the ‘optimal TMDI’, constitute the ADAS in the present study. This is merely a comparison; additional optimal ADAS exploration can be conducted to achieve the desired response control of the base-isolated structures.

A comparison of the normalized displacement variance of the base-isolated structure with the optimal TMDI and corresponding ADAS is depicted in Figure 4. The plotting of the response is done for the damping ratio, ${\xi }_b$ = 0.05, 0.1, 0.15, and 0.2, and the isolation period, T_b = 3 s. The figure reveals that for the lower inertance ratios, the normalized variance of base-isolated TMDI is lower than the corresponding ADAS. However, for the higher inertance ratio, the ADAS is superior to the TMDI in reducing the displacement variance of the base-isolated structure. This can be explained by the fact that while the TMDI’s performance is unaffected by higher inertance ratios, the amount of auxiliary damping (i.e., c_t) needed to achieve the best results is higher [40]. The performance of the isolated structure in displacement reduction improves when this damper with high damping as ADAS is attached directly to the ground and isolation floor. This figure also indicates that the reduction in displacement by the TMDI is more significant for isolated structures with lower damping in comparison to higher damping and confirms the recent finding [40].

3. Rigid Base-Isolated Structure with FID and FD

Figure 5a depicts a schematic diagram of FID installed in a rigid model of the base-isolated structure. The two terminals, represented by 1 and 2 on the FID, are coupled with the ends depicted in Figure 1a of the base-isolated structure model. An FID typically consists of a piston and cylinder that force the fluid through a helical tube that is encircled by the cylinder. The hydraulic cylinder and the piston rod are the two terminals of the FID. The fluid flow through an external helical channel produces rotational inertia to compensate for the pressure loss [42,43,44]. The FID’s resisting force depends on the friction, oil density, and viscosity of the fluid. The FID is modeled as a linear inerter in parallel with a linear [27,28,29,32,33] and a non-linear [30,31] dashpot for studying the seismic response of the base-isolated structures. For simplicity, the FID is modeled in the present study as a linear inerter (i.e., with an inertance coefficient of b) in parallel with a linear viscous FD (i.e., with a damping coefficient of c_f). It is worth noting that this FID modeling is identical to the EIMD modeling described in the References [35,36].

The FID is defined by two parameters, β and ξ_f, which are both defined as follows:

$$\beta =\frac{b}{m} \mathrm{and} {\xi }_f=\frac{c_f}{2m{\omega }_b}$$

(10)

Consider the rigid superstructure model of the base-isolated structure (refer to Figure 1a) with supplemental FID. The governing equation of motion of a base-isolated structure with FID under earthquake excitation is expressed as

$$(m+b)\ddot{x}+(c_b+c_f)\dot{x}+k_{b}x=-m{\ddot{x}}_g$$

(11)

Dividing Equation (11) by (m + b) and substituting $\beta =b/m$, the equation of motion reduces to

$$\ddot{x}+2\left(\frac{{\xi }_b+{\xi }_f}{\sqrt{1+\beta }}\right)\left(\frac{{\omega }_b}{\sqrt{1+\beta }}\right)\dot{x}+{\left(\frac{{\omega }_b}{\sqrt{1+\beta }}\right)}^{2}x=-\left(\frac{1}{1+\beta }\right){\ddot{x}}_g$$

(12)

The above indicates that FID elongates the vibration period and decreases the total damping ratio (i.e., ${\xi }_b+{\xi }_f$) of the base-isolated structure. It also suppresses the level of earthquake shaking (as the denominator on the right-hand side is always larger than unity).

Let the rigid superstructure model of the base-isolated structure with FID be subjected to white-noise earthquake acceleration with PSDF as S₀. The mean square displacement of the isolated structure with FID will be as follows [45]:

$${\sigma }_x^2=\frac{\pi S_0}{2({\xi }_b+{\xi }_f){\omega }_b^3}$$

(13)

The above equation indicates that for broadband excitation, the displacement variance of the base-isolated structure is not influenced by the inertance of the FID. The reduction in displacement is primarily happening due to the added damping of the FID. Since the inerter of the FID does not influence the response of the isolated structures under broad band-excitation, it is therefore proposed that the corresponding FD only be added parallel to the isolation system (refer to Figure 5b).

The absolute acceleration, ${\ddot{x}}_a$, of the isolated structural mass with FID, is expressed as

$${\ddot{x}}_a=[\ddot{x}+{\ddot{x}}_g]=-[{\omega }_b^{2}x+2({\xi }_b+{\xi }_f){\omega }_b\dot{x}+\beta \ddot{x}]$$

(14)

Interestingly, two terms, i.e., $2{\xi }_f{\omega }_b\dot{x}$ and $\beta \ddot{x}$, are added to the absolute acceleration of the mass compared to the conventional isolation system, and their effects are interesting.

When subjected to the harmonic earthquake acceleration as ${\ddot{x}}_g=e^{j\omega t}$, the steady-state absolute acceleration, ${\ddot{x}}_a$, is expressed as

$${\ddot{x}}_a=H_{{\ddot{x}}_a}\left(j\omega \right)e^{j\omega t}$$

(15)

where $H_{{\ddot{x}}_a}\left(j\omega \right)$ is the FRF of the response ${\ddot{x}}_a$.

In Figure 6, the variation of the FRF of the absolute acceleration of the mass of the isolated structure with FID is plotted. The FRF with β = 0 corresponds to the structure without FID. Because of the added supplement inertial mass by the FID, the resonating frequency decreases as the inertance ratio increases. The peak values of absolute acceleration also decrease with the increase in β. However, for higher frequencies, the amplitude of absolute acceleration increases with β and remains constant throughout. This implies that FID will transmit earthquake vibrations into the base-isolated structures associated with the higher frequencies. This can be detrimental to the sensitive high-frequency equipment or secondary system installed in the isolated structure. It is also to be noted that the mean square absolute acceleration of the isolated structure with FID, ${\sigma }_{{\ddot{x}}_a}^2\to \infty$ when subjected to white-noise earthquake excitation.

4. Response under Filtered White-Noise Excitation

Earthquakes are characterized by an unpredictable and multi-dimensional nature. When the evolution of frequency content is overlooked, the stationary nature of ground motion can be expressed through a power spectral density function (PSDF) matrix. The PSDF for the earthquake excitation is assumed to align with the model proposed by Kanai and Tajimi [46,47], i.e.,

$$S_{{\ddot{x}}_g}(\omega )=S_0\left(\frac{1+4{\xi }_g^2(\omega /{\omega }_g{)}^2}{[1-(\omega /{\omega }_g{)}^2{]}^2+4{\xi }_g^2(\omega /{\omega }_g{)}^2}\right)$$

(16)

where $S_{{\ddot{x}}_g}(\omega )$ = PSDF of the earthquake acceleration ${\ddot{x}}_g$; ω_g = predominant frequency of the soil media or filter; and ${\xi }_g$ = damping ratio of the soil media or filter. The chosen parameters for filter media are ω_g = 15 rad/s and ${\xi }_g=0$.6, which represents the firm types of soil strata.

A comparison of ${\tilde{\sigma }}_x^2$ of the base-isolated with optimal TMDI, ADAS, FID, and FD subjected to the Kanai–Tajimi model of earthquake excitation is shown in Figure 7. The responses are plotted for the isolation period, T_b = 3 and 4 s, and the damping ratio, ${\xi }_b$ = 0.1. As observed earlier, the performance of the corresponding ADAS device is better than the TMDI for higher inertance ratios. The displacement variance of the base-isolated structure with FID and corresponding FD under filtered white-noise excitation also does not differ, as was also seen for the white-noise excitation.

5. Flexible Isolated Structure under Real Earthquakes

An earlier study attempted the seismic response of a rigid base-isolated structure with the optimum TMDI, ADAS, FID, and FD under a stationary earthquake ground excitation model. However, the behavior of these devices under real earthquake excitation and when the isolated superstructure is assumed to be flexible will be interesting to explore. This will provide insight into the tuning behavior and the frequency contents of the superstructure acceleration of the isolated structure with TMDI and FID. Figure 8a depicts a typical five-story linear shear-type building equipped with supplementary TMDI, ADAS, FID, and FD devices in addition to a base isolation system. The base mass and the ith floor mass are represented by $m_b$ and $m_i$, respectively. $k_i$ stands for the ith floor’s stiffness. The lead rubber bearing (LRB) system is selected as an isolation device. Hysteretic damping is additionally produced by the lead core of the LRB through yielding, which dissipates the seismic energy input. Figure 8b illustrates the simple bi-linear force-deformation behavior of the LRB. The bi-linear characteristics of the LRB (lead rubber bearing) are defined by three parameters: F_y, (yield force), k_b (post-yield stiffness), and q (yield displacement). For the selected base-isolated structure with supplemental TMDI, ADAS, FID, and FD, the governing equations of motion are

$$\mathbf{M}\ddot{\mathrm{x}}\left(t\right)+\mathbf{C}\dot{\mathrm{x}}\left(t\right)+\mathbf{K}\mathrm{x}\left(\mathrm{t}\right)+{\mathbf{D}}_i{F}_i\left(t\right)+{\mathbf{D}}_s{\mathbf{F}}_{\mathbf{s}}\left(t\right)=-\mathbf{E}{\ddot{x}}_g\left(t\right)$$

(17)

where M = mass matrix; C = damping matrix; K = stiffness matrix; x(t) = vector of horizontal displacements (relative to the ground) of each mass at time t; D_i = location vector for the bi-linear restoring force of the LRB; F_i(t) = bi-linear restoring force of the LRB; D_s = location matrix for the vector of control forces F_s(t) produced by the supplemental devices TMDI, ADAS, FID, and FD; E = vector containing the vibrating masses; and ${\ddot{x}}_g\left(t\right)$ = earthquake acceleration specified by a time history.

The study focuses on a five-story building characterized by linear inter-story stiffness and viscous damping, with parameters sourced from Kelly [48]. The mass distribution is uniform across all floors and the base raft locations, implying equal masses (m_b = m_i for i = 1 to 5). The inter-story stiffness for each floor is denoted as k₁, k₂, k₃, k₄, and k₅, corresponding to 15, 14, 12, 9, and 5 times the k, respectively. Here, the value for k is chosen such that it gives a fundamental time period of 0.4 s for the fixed-base structure. The corresponding fixed base structure has five natural frequencies, quantified in radians per second: 15.71, 38.48, 60.84, 83.12, and 105.37. With the assumption that the modal damping ratio for all vibrational modes is 2 percent, the damping matrix for the superstructure is formulated. The parameters T_b and ξ_b (see Equation (1)) are used to calculate the post-yield stiffness and damping of the LRB while considering the flexible base-isolated structure’s total mass, M (i.e., $M=m_b+{\sum }_{i=1}^5{m}_i$). The isolation damping ratio is chosen as 0.1 of the critical damping. The yield strength of the LRB is selected as 0.05W (i.e., W = Mg represents the total weight of the base-isolated structure) and the yield displacement as 0.025 m. The normalized inertance, β of the supplemental devices is calculated by considering the total mass, M. The auxiliary stiffness and damping of the optimal TMDI (for the specified value of T_b and ${\xi }_b$) are selected corresponding to which minimizes the ${\sigma }_x^2$ of a rigid-base isolated structure under white-noise earthquake excitation using Equations (8) and (9). The viscous damping of FID is defined by ${\xi }_f$ using Equation (10) and considering the total mass of the isolated structure, and taken as ${\xi }_f=\mathsf{\beta }$/2. The analytical seismic response of the isolated structure with supplemental TMDI, ADAS, FID, and FD devices is evaluated numerically by solving the governing equations of motion using the step-by-step method [49].

Three real earthquake motions, namely the north–south component of El-Centro, 1940 earthquake, the N00E component of the 1989 Loma Prieta earthquake (measured at Los Gatos Presentation Centre), and the 270 component of the 1992 Landers earthquake (measured at Lucerne Valley), are selected. The peak ground acceleration (PGA) of El-Centro, Loma Prieta, and Landers earthquake motions are 0.34 g, 0.57 g, and 0.73 g, respectively. The response quantities of interest are the top floor absolute acceleration (${\ddot{x}}_5^a$), relative base displacement (x_b), force in the isolation system (F_is), force in the supplemental device (F_sup), and total base shear (F). The selected force quantities are normalized with W.

The time history of the top floor/story acceleration, relative base displacement, the force developed in the isolator, the force developed in the supplemental control system used, and total base shear are shown in Figure 9. The responses are shown for the Loma Prieta, 1989 earthquake, and they are compared with the corresponding uncontrolled response of isolated structures, i.e., without supplemental devices (referred to as BIS). The figure indicates that the peak value of the top floor acceleration for BIS + TMDI, BIS + ADAS, and BIS + FD is the same as that of BIS, suggesting that these supplemental control devices are not much altering the top floor acceleration. However, there is an almost 3.61 times increase in the peak top floor acceleration of the isolated structure with FID compared to the corresponding peak acceleration of BIS. This is unacceptable as the FID defeats the primary purpose of seismic isolation by increasing the structural acceleration. It was also noted that the base-isolated structure with FID’s absolute acceleration contains high-frequency components, which could be harmful to equipment or a secondary system that is sensitive to high-frequency vibrations. There is a reduction in base displacement by all the supplemental devices. It was observed to decrease by 23.8 to 32.8 percent compared to the corresponding BIS. The base displacement of isolated structures with ADAS is less than the corresponding TMDI. This is interesting and primarily happens because the peak base displacement occurs in the large initial cycles of the response. By that time, the TMDI was not adequately tuned to dissipate the input seismic energy. However, the TMDI is more effective in reducing the subsequent peak response in comparison to the ADAS. The time variation in force within the isolation system mirrors the displacement of the isolator for all supplementary devices.

The effects of the β on the peak top floor absolute acceleration, relative base displacement, the force in the isolation system, the force in the supplemental device, and the total base shear of the base-isolated structure with different supplemental devices and earthquakes are shown in Figure 10. The figure indicates that the peak top floor acceleration is not very sensitive to β and quite comparable to the base-isolated structures with supplemental TMDI, ADAS, and FD. However, the peak top floor acceleration increases with β for the isolated structure with supplemental FID. This can be explained with the help of Figure 6, which shows that the grounded inerter connected to the base mass will directly transmit the vibration (i.e., accelerations) associated with high-frequency amplitudes in the earthquake ground motion. This affirms that the inclusion of an inerter in the Fluid Inerter Damper (FID) as a supplementary device to the isolation system has adverse consequences, leading to raised structural accelerations. As β increases, the maximum displacement at the base of the base-isolated structure diminishes when supplemented with devices. A contrast of the peak base displacement of the base-isolated structures with TMDI and the corresponding ADAS indicates a greater reduction in the base displacement by ADAS compared to TMDI.

Further, there is a greater reduction in base displacement for the isolated structures with FID than with FD. The variation of peak force in the isolation system has a similar trend to base displacement. The peak force in the supplemental devices increases with the increase in β. However, the force in the FID is substantially higher in comparison to the force of other supplemental devices. The peak total base shear of isolated structures with TMDI, ADAS, and FD is not very sensitive to β, but it increases for the FID. Thus, it can be concluded that the inerter in the FID increases the structural acceleration as well as the total base shear.

The effects of β on the peak top floor absolute acceleration, relative base displacement, the force in the isolation system, the force in the supplemental device, and the total base shear of the base-isolated structures with T_b = 4 s for different earthquakes are shown in Figure 11. The trends of the results in Figure 11 are similar to those observed in Figure 10 for considering T_b = 3 s.

Based on the discussion of the presented results, it can be concluded that there is a greater or comparable reduction in the base displacement by the corresponding ADAS in comparison to the optimal TMDI. Further, an inerter of the FID attached to the ground and an isolated structure have detrimental effects on seismic isolation by transmitting the vibrations associated with high-frequency accelerations and increasing the base shear. The study’s overall conclusion is that inerter-based supplemental devices such as TMDI and FID may not be very effective for the base-isolated structures to warrant their practical application. On the other hand, a pair of inerters with a clutching effect might be able to significantly reduce the displacement in the base-isolated structure’s isolation system [50,51,52,53]. Also, the performance of the inerter-based dampers in conjunction with negative stiffness devices was somewhat encouraging for the base-isolated structures [54,55]. The current study is primarily concerned with the inerter’s function in the recently investigated supplemental dampers for base-isolated structures. However, it is strongly recommended to compare their performance with the widely researched dampers, such as magnetorheological dampers, negative stiffness dampers, semi-active variable friction and stiffness devices, etc. It is also recommended that the findings of the present study be experimentally validated.

6. Conclusions

From this research study of base-isolated structures equipped with inerter-based supplemental devices, the following conclusions can be drawn:

• The TMDI (especially with a sizeable inertance ratio) may not control the response of the base-isolated structure through tuning effects but is controlled by the added heavy damping and stiffness of its auxiliary damper under real earthquake excitation. It is recommended that the corresponding damping and stiffness (i.e., ADAS) of the TMDI be installed in parallel to the isolation device for better response control and economy.
• The variance of base-isolated structures with optimal TMDI subjected to broadband earthquake excitation is higher than the corresponding ADAS for lower values of the inertance ratio. However, for the higher inertance ratio, the ADAS is superior to the TMDI in reducing the displacement variance of the base-isolated structures.
• For a broadband excitation, the displacement variance of the base-isolated structure with supplemental FID is not influenced by the inerter of the FID. The reduction in displacement is primarily due to the added damping of the FID. It is recommended that the FD alone can better control the response of the base-isolated structure as compared to the FID.
• There is a greater or comparable reduction in the base displacement of the isolated structures under real earthquake excitation by ADAS compared to the corresponding optimal TMDI. It implies that with a suitable ADAS, better control of response in the isolation system can be achieved.
• The absolute structural acceleration of a base-isolated structure with a fluid inerter damper (FID) exhibits the presence of high-frequency components, potentially posing adverse impacts on equipment or secondary systems sensitive to high-frequency disturbances. An inerter attached to the ground and an isolated structure defeat the purpose of seismic isolation by transmitting the vibrations and increasing the base shear of the structure.
• The inerter-based supplemental devices such as TMDI (or TID) and FID (or EIMD) may not be very effective for the base-isolated structures to warrant their practical application.